1. 17 A – Cubic Polynomials 1: Graphing Basic Cubic Polynomials

2. Cubic Polynomials A cubic polynomial is a degree 3 polynomial in the form f(x) = ax 3 + bx 2 + cx + d.

3. Volume A 40 cm by 30 cm sheet of tinplate is to be used to make a cake tin. Squares are cut from its corners and the metal is then folded upwards along the dashed lines. Edges are fixed together to form the open rectangular tin. Consequently the capacity of the cake tin V, is given by V(x) = x(40 – 2x)(30 – 2x).  How does the capacity change as x changes?  What are the restrictions on x?  What sized squares must be cut out for the cake tin to have maximum capacity?

4. Forms of Cubics The function V(x) = x(40 – 2x)(30 – 2x) is the factored form of a cubic polynomial. The expanded form (or standard form) can be found by multiplying the factored form.  V(x) = 4x 3 – 140x 2 + 1200x  This form allows you to see why this function is considered a cubic polynomial.

5. Expanding Cubics Write y = 2(x – 1) 3 + 4 in general form (expand).

6.  Write f(x) = 2(x – 3) 3 + 7 in general form (expand).

7. Graphing Cubics Use technology to assist you to draw sketch graphs of: f(x) = x 3 f(x) = -x 3 f(x) = 2x 3 f(x) = ½x 3  What effect does a have in f(x) = ax 3 ?

8. Graphing Cubics Use technology to assist you to draw sketch graphs of: f(x) = x 3 f(x) = x 3 + 2 f(x) = x 3 – 3  What effect does k have in f(x) = x 3 + k?

9. Graphing Cubics Use technology to assist you to draw sketch graphs of: f(x) = x 3 f(x) = (x + 2) 3 f(x) = (x – 3) 3  What effect does h have in f(x) = (x – h) 3 ?

10. Graphing Cubics Use technology to assist you to draw sketch graphs of: f(x) = (x – 1) 3 + 2 f(x) = (x + 2) 3 + 1  What is important about (h, k) in f(x) = (x – h) 3 + k?